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Approximate earth mover’s distance in linear time
"... The earth mover’s distance (EMD) [16] is an important perceptually meaningful metric for comparing histograms, butitsuffersfromhigh(O(N 3 log N))computationalcomplexity. We present a novel linear time algorithm for approximating the EMD for low dimensional histograms using the sum of absolute values ..."
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The earth mover’s distance (EMD) [16] is an important perceptually meaningful metric for comparing histograms, butitsuffersfromhigh(O(N 3 log N))computationalcomplexity. We present a novel linear time algorithm for approximating the EMD for low dimensional histograms using the sum of absolute values of the weighted wavelet coefficients of the difference histogram. EMD computation is aspecialcaseoftheKantorovichRubinsteintransshipment problem, andweexploittheHölder continuityconstraintin its dual form to convert it into a simple optimization problem with an explicit solution in the wavelet domain. We prove that the resulting wavelet EMD metric is equivalent to EMD, i.e. the ratio of the two is bounded. We also provide estimates for the bounds. Theweightedwavelettransformcanbecomputedintime linear in the number of histogram bins, while the comparison is about as fast as for normal Euclidean distance or χ 2 statistic. WeexperimentallyshowthatwaveletEMDisa good approximation to EMD, has similar performance, but requires much less computation. 1.
Local Histogram based Segmentation using the Wasserstein Distance
, 2008
"... We propose and analyze a nonparametric regionbased active contour model for segmenting cluttered scenes. The proposed model is unsupervised and assumes that pixel intensity is independently identically distributed. The proposed energy functional consists of a geometric regularization term that pena ..."
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Cited by 25 (0 self)
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We propose and analyze a nonparametric regionbased active contour model for segmenting cluttered scenes. The proposed model is unsupervised and assumes that pixel intensity is independently identically distributed. The proposed energy functional consists of a geometric regularization term that penalizes the length of region boundaries, and a regionbased image term that uses the probability density function (or histogram) of pixel intensity to distinguish different regions. More specifically, the region data encourages partitioning the image domain so that the local histograms within each region are approximately homogeneous. The solutions of the proposed model do not need to differentiate histograms. The similarity between normalized histograms is measured by the Wasserstein distance with exponent 1, which is able to fairly compare two histograms, both continuous and discontinuous. We employ a fast global minimization method based on [11, 6] to solve the proposed model. The advantages of this method include less computational time compared with the minimization method by gradient descent of the associated EulerLagrange equation [12] and the abil
An Image Morphing Technique Based on Optimal Mass Preserving Mapping
 IEEE Transactions on Image Processing
, 2007
"... Abstract—Image morphing, or image interpolation in the time domain, deals with the metamorphosis of one image into another. In this paper, a new class of image morphing algorithms is proposed based on the theory of optimal mass transport. The 2 mass moving energy functional is modified by adding an ..."
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Cited by 14 (0 self)
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Abstract—Image morphing, or image interpolation in the time domain, deals with the metamorphosis of one image into another. In this paper, a new class of image morphing algorithms is proposed based on the theory of optimal mass transport. The 2 mass moving energy functional is modified by adding an intensity penalizing term, in order to reduce the undesired double exposure effect. It is an intensitybased approach and, thus, is parameter free. The optimal warping function is computed using an iterative gradient descent approach. This proposed morphing method is also extended to doubly connected domains using a harmonic parameterization technique, along with finiteelement methods. Index Terms—Image interpolation, image morphing, image warping, mass preserving mapping, Monge–Kantorovich flow, optimal transport. I.
Texture Mapping via Optimal Mass Transport
, 2010
"... In this paper, we present a novel method for texture mapping of closed surfaces. Our method is based on the technique of optimal mass transport (also known as the “earthmover’s metric”). This is a classical problem that concerns determining the optimal way, in the sense of minimal transportation co ..."
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Cited by 12 (0 self)
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In this paper, we present a novel method for texture mapping of closed surfaces. Our method is based on the technique of optimal mass transport (also known as the “earthmover’s metric”). This is a classical problem that concerns determining the optimal way, in the sense of minimal transportation cost, of moving a pile of soil from one site to another. In our context, the resulting mapping is area preserving and minimizes angle distortion in the optimal mass sense. Indeed, we first begin with an anglepreserving mapping (which may greatly distort area) and then correct it using the mass transport procedure derived via a certain gradient flow. In order to obtain fast convergence to the optimal mapping, we incorporate a multiresolution scheme into our flow. We also use ideas from discrete exterior calculus in our computations.
AN EFFICIENT NUMERICAL METHOD FOR THE SOLUTION OF THE L2 OPTIMAL MASS TRANSFER PROBLEM
, 2010
"... In this paper we present a new computationally efficient numerical scheme for the minimizing flow approach for the computation of the optimal L2 mass transport mapping. In contrast to the integration of a time dependent partial differential equation proposed in [S. Angenent, S. Haker, and A. Tannen ..."
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In this paper we present a new computationally efficient numerical scheme for the minimizing flow approach for the computation of the optimal L2 mass transport mapping. In contrast to the integration of a time dependent partial differential equation proposed in [S. Angenent, S. Haker, and A. Tannenbaum, SIAM J. Math. Anal., 35 (2003), pp. 61–97], we employ in the present work a direct variational method. The efficacy of the approach is demonstrated on both real and synthetic data.
FIVE LECTURES ON OPTIMAL TRANSPORTATION: GEOMETRY, REGULARITY AND APPLICATIONS
, 2010
"... In this series of lectures we introduce the MongeKantorovich problem of optimally transporting one distribution of mass onto another, where optimality is measured against a cost function c(x, y). Connections to geometry, inequalities, and partial differential equations will be discussed, focusing i ..."
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Cited by 12 (4 self)
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In this series of lectures we introduce the MongeKantorovich problem of optimally transporting one distribution of mass onto another, where optimality is measured against a cost function c(x, y). Connections to geometry, inequalities, and partial differential equations will be discussed, focusing in particular on recent developments in the regularity theory for MongeAmpère type equations. An application to microeconomics will also be described, which amounts to finding the equilibrium price distribution for a monopolist marketing a multidimensional line of products to a population of anonymous agents whose preferences are known only statistically.
Histogram Based Segmentation Using Wasserstein Distances
"... Abstract. In this paper, we propose a new nonparametric regionbased active contour model for clutter image segmentation. To quantify the similarity between two clutter regions, we propose to compare their respective histograms using the Wasserstein distance. Our first segmentation model is based on ..."
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Abstract. In this paper, we propose a new nonparametric regionbased active contour model for clutter image segmentation. To quantify the similarity between two clutter regions, we propose to compare their respective histograms using the Wasserstein distance. Our first segmentation model is based on minimizing the Wasserstein distance between the object (resp. background) histogram and the object (resp. background) reference histogram, together with a geometric regularization term that penalizes complicated region boundaries. The minimization is achieved by computing the gradient of the level set formulation for the energy. Our second model does not require reference histograms and assumes that the image can be partitioned into two regions in each of which the local histograms are similar everywhere. Key words: image segmentation, regionbased active contour, Wasserstein distance, clutter 1
OPTIMAL TRANSPORT WITH PROXIMAL SPLITTING
"... Abstract. This article reviews the use of first order convex optimization schemes to solve the discretized dynamic optimal transport problem, initially proposed by Benamou and Brenier. We develop a staggered grid discretization that is well adapted to the computation of the L2 optimal transport geod ..."
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Abstract. This article reviews the use of first order convex optimization schemes to solve the discretized dynamic optimal transport problem, initially proposed by Benamou and Brenier. We develop a staggered grid discretization that is well adapted to the computation of the L2 optimal transport geodesic between distributions defined on a uniform spatial grid. We show how proximal splitting schemes can be used to solve the resulting large scale convex optimization problem. A specific instantiation of this method on a centered grid corresponds to the initial algorithm developed by Benamou and Brenier. We also show how more general cost functions can be taken into account and how to extend the method to perform optimal transport on a Riemannian manifold. hal00816211, version 1 21 Apr 2013 1. Introduction. Optimal
Mathematical Methods in Medical Image Processing
 BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY
, 2006
"... In this paper, we describe some central mathematical problems in medical imaging. The subject has been undergoing rapid changes driven by better hardware and software. Much of the software is based on novel methods utilizing geometric partial di#erential equations in conjunction with standard si ..."
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Cited by 8 (0 self)
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In this paper, we describe some central mathematical problems in medical imaging. The subject has been undergoing rapid changes driven by better hardware and software. Much of the software is based on novel methods utilizing geometric partial di#erential equations in conjunction with standard signal/image processing techniques as well as computer graphics facilitating man/machine interactions. As part of this enterprise, researchers have been trying to base biomedical engineering principles on rigorous mathematical foundations for the development of software methods to be integrated into complete therapy delivery systems. These systems support the more effective delivery of many imageguided procedures such as radiation therapy, biopsy, and minimally invasive surgery. We will show how mathematics may impact some of the main problems in this area including image enhancement, registration, and segmentation.
Brain surface warping via minimizing Lipschitz extensions
 in Proc. Workshop Math. Foundations Computational Anatomy (MFCA
"... Based on the notion Minimizing Lipschitz Extensions and its connection with the infinity Laplacian, a computational framework for surface warping and in particular brain warping (the nonlinear registration of brain imaging data) is presented in this paper. The basic concept is to compute a map betwe ..."
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Cited by 7 (2 self)
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Based on the notion Minimizing Lipschitz Extensions and its connection with the infinity Laplacian, a computational framework for surface warping and in particular brain warping (the nonlinear registration of brain imaging data) is presented in this paper. The basic concept is to compute a map between surfaces that minimizes a distortion measure based on geodesic distances while respecting the boundary conditions provided. In particular, the global Lipschitz constant of the map is minimized. This framework allows generic boundary conditions to be applied and allows direct surfacetosurface warping. It avoids the need for intermediate maps that flatten the surface onto the plane or sphere, as is commonly done in the literature on surfacebased nonrigid brain image registration. The presentation of the framework is complemented with examples on synthetic geometric phantoms and cortical surfaces extracted from human brain MRI scans. 1